McWeeny procedure

We will now discuss an iterative scheme based on the CHF approach outlined in Sections 11 and 111, using the McWeeny procedure [7] for resolving matrices into components, by introducing projection operators R and R2, with respect to the subspaces spanned by occupied and virtual molecular orbitals. 

A lot of effort has gone into devising procedures for solving the problem. You might like to read about direct procedures for finding P in McWeeny and Sutcliffe (1969). Roughly, what we do is this ... 

It is still possible to introduce further approximations, in the form of a total neglect of the two-electron interactions. This means that the matrix elements are given in terms only of the one-electron integrals Huu and Huv, respectively. In such a case, as the matrix elements are independent of the molecular orbitals, no (trial and error) iteration procedure is needed. Diagonalization of the matrix yields directly both the eigenvectors and eigenvalues. It should be mentioned, however, that McWeeny (1964) has developed a SCF formulation of this theory. 

Efficient techniques for optimizing orbitals have been elaborated in multiconfiguration SCF (MCSCF) theory (see e.g. the book by McWeeny [134] and refs, therein). Since the APSG wave function represents a special class among MCSCF functions, these procedures can be applied to determine the optimal Arai-subspaces [65],... 

Once the momentum wave function is given, the momentum density p(p) and the other reduced density matrices in momentum space are obtained by the same procedure as in position space (Lowdin, 1955 McWeeny, 1960 Davidson, 1976). For example, the momentum density is given by... 

When dealing with relativistic hamiltonians, a great amount of new operators appears because of several approximate procedures allowing the simplification of Dirac s equation. See, for example, the treatise of Bethe and Salpeter [81], as well as the Moss discussion [77 a)], the McWeeny s book [82], or a recent comprehensible brief resume due to Almldf and Gropen [83]. 

The procedure [due to Ramsey (see also McWeeny )] for calculating o is to introduce the complete vector potential including that due to the nuclear magnetic moments, M r, of each nucleus. 

However, using the McWeeny approach [7], it is sufficient to calculate only the projection P >v/K on the subspace of virtual zero-order orbitals in order to get the second hyperpolarizability tensor. This projection is evaluated via a procedure similar to the one used in solving the first-order equation (21). Taking in (32) the Hermitian product with the unoccupied and using (19), one finds... 

At this stage it is, however, not clear which of these parametrizations should be preferred over the others for application in decoupling procedures and whether they all yield identical block-diagonal Hamiltonians. Furthermore, the four possibilities given above to parametrize unitary transformations differ obviously in their radius of convergence Rc, which is equal to unity for the square root and the McWeeny form, whereas Rc = 2 for the Cayley parametrization, and Rc = co for the exponential form. 

The natural-orbital method must be used with care for states that are not totally symmetric in space and spin, since the natural orbitals will not then be symmetry-adapted (see e.g. McWeeny and Kutzelnigg, 1968). In such cases symmetry-adapted orbitals may be obtained by diagonalizing the totally symmetric projection of the density matrix as discussed by Ruedenberg et al. (1979). In general, the procedure appears to be rather efficient. 

In the derivation of density matrix-based SCF theory below, we do not employ the chemical potential introduced by LNV, but instead we follow the derivation of Ochsenfeld and Head-Gordon, because McWeeny s purification automatically preserves the electron number.Therefore, to avoid the diagonalization within the SCF procedure, we minimize the energy functional... 

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